injective hull

Equivalently, Q is an injective hull of X
if Q is injective,
and X is a submodule of Q,
and if g:X→Q′ is a monomorphism
from X to an injective module Q′,
then there exists a monomorphism h:Q→Q′
such that h⁢(x)=g⁢(x) for all x∈X.

\xymatrix&0\ar[d]0\ar[r]&X\ar[r]i\ar[d]g&Q\ar@-->[dl]h&Q′

Every module X has an injective hull, which is unique up to isomorphism. The injective hull of X is sometimes denoted E⁢(X).